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In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalence relation.A ternary equivalence relation is symmetric, reflexive, and transitive, where those terms are meant in the sense defined below.
A reflexive and symmetric relation is a dependency relation (if finite), and a tolerance relation if infinite. A preorder is reflexive and transitive. A congruence relation is an equivalence relation whose domain X {\displaystyle X} is also the underlying set for an algebraic structure , and which respects the additional structure.
This last property is often called co-transitivity or comparison. The complement of an apartness relation is an equivalence relation, as the above three conditions become reflexivity, symmetry, and transitivity. If this equivalence relation is in fact equality, then the apartness relation is called tight.
However, there is a formula for finding the number of relations that are simultaneously reflexive, symmetric, and transitive – in other words, equivalence relations – (sequence A000110 in the OEIS), those that are symmetric and transitive, those that are symmetric, transitive, and antisymmetric, and those that are total, transitive, and ...
An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.
To preserve transitivity, one must take the transitive closure. This occurs, for example, when taking the union of two equivalence relations or two preorders. To obtain a new equivalence relation or preorder one must take the transitive closure (reflexivity and symmetry—in the case of equivalence relations—are automatic).
Reflexivity For every a, one has a = a. Symmetry For every a and b, if a = b, then b = a. Transitivity For every a, b, and c, if a = b and b = c, then a = c. [11] [12] Substitution Informally, this just means that if a = b, then a can replace b in any mathematical expression or formula without changing its meaning. (For a formal explanation ...
S4 := reflexive and transitive; S5 := reflexive and Euclidean; The Euclidean property along with reflexivity yields symmetry and transitivity. (The Euclidean property can be obtained, as well, from symmetry and transitivity.) Hence if the accessibility relation R is reflexive and Euclidean, R is provably symmetric and transitive as well.